Robust Adaptive Beamforming With Nonconvex Union of Multiple Steering Vector Uncertainty Sets

This paper addresses a robust adaptive beamforming (RAB) problem by maximizing the worst-case signal-to-interference-plus-noise ratio (SINR) over a union of small uncertainty sets, each including a similarity constraint on the desired signal steering vector. To capture uncertainty more comprehensively than a single large sphere, the union of small sets is employed, allowing improved adaptability. The RAB problem is reformulated as a minimization of a convex quadratic objective under constraints formed by the difference of convex quadratic functions. Then, a sequential second-order cone programming (SOCP) approximation algorithm is proposed with reduced computational cost and enhanced beamformer output SINR compared to existing methods. The algorithm generates a nonincreasing, bounded sequence of SOCP optimal values, ensuring the sequence's convergence to a locally optimal RAB solution. Further, each uncertainty set is extended with one more norm constraint, reflecting practical array configurations, and the SINR maximization problem is converted into a quadratic matrix inequality (QMI) problem. A rank-reduction technique is applied to obtain a rank-one solution for the linear matrix inequality relaxation problem of it. Additionally, for the minimum variance distortionless response (MVDR) RAB problem with the nonconvex union of the extended uncertainty sets, an algorithm is developed to solve homogeneous quadratically constrained quadratic programming (QCQP) subproblems, and an optimal beamformer for the MVDR RAB problem is obtained by selecting the best solution (among all globally optimal QCQP solutions) corresponding to the maximal array output power. Simulation results confirm the proposed beamformers' superiority in output SINR, computational efficiency, and normalized beampattern compared to existing methods.